thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X , Y ;
Y c= Z ;
A // M ;
let U , S ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G , H ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X , Y ;
let C ;
x _|_ p ;
o is monotone ;
let X , Y ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `1 = b `1 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b , c be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = - 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R . x <> 0 ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Point of TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume not F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in dom f ;
cluster uparrow x -> \mathclose { \rm c } ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
L~ 0. D >= G2 ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded upper-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
and Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pare c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
C c= dom f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing implies seq is non-decreasing
Gseq is non-decreasing implies seq is non-decreasing
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function of S , X ;
assume P [ n ] ;
assume union S is finite independent finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , f be Function of s , I ;
b ` ` c= b9 ` ;
assume not x in mSet ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an1 < n & n <= len bn1 ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat , x be Element of REAL ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s , k ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , a be Element of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of X , REAL ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ` ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> preFunctor ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co /\ L~ co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be void mamaid id id ;
let N be non empty `2 \in \in the non empty Subset of M ;
let R be RelStr with finite is finite ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as Int-Location ;
assume not I does not ` ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> ManySortedSet of I ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> G2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A3 & A2 <> C ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec * f ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 + 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 & 1 in dom D2 ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( for i being Nat holds i in i implies ( i in dom f ) ) implies ( i in dom f )
X1 c= dom f & X1 c= dom g ;
h . x in h . a ;
let G be e in the space of \cap \mathopen { \rbrack } 0 , 1 .[ ;
cluster m * n -> square ;
let k9 be Nat , n be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 ^ <* p *> ) ;
p is FinSequence of X & q is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BX c= XX \/ XY ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 `2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be K of X , M be Element of S ;
i . y in rng i ;
R^1 c= dom f & dom g c= dom f ;
f . x in rng f ;
mt <= ( r - 1 ) / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F | A ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , x be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n1 + 1 < n2 + 1 ;
n1 < n1 + 1 & n1 + 1 < n2 + 1 ;
cluster [: T , X :] -> \overline ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT , n ;
let S be Subset of R ;
y in rng S29 & y in rng S29 ;
b = sup dom f & b = sup dom g ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 * h2 ) ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT , j be Element of NAT ;
Support u = Support p \/ Support q ;
assume X is complete `2 ;
assume f = g & p = q ;
n1 <= n1 + 1 - 1 ;
let x be Element of REAL , r be Real ;
assume x in rng ( s2 - s1 ) ;
x0 < x0 + 1 / ( n + 1 ) ;
len ( Carrier ( L ) ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT , n ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be Element of NAT ;
assume z in x := being being being being being being being such that z in x := y ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n + m ) = n ;
\mathop { Z } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E |^ omega ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} ;
L1 /\ L2 = {} implies L1 is open
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-129 & x in rng f-129 ;
set nn8 = n + j ;
let D7 be non empty set , f be FinSequence of D , i be Nat ;
let K be right_zeroed non empty addLoopStr , p be Polynomial of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is total implies R1 + R2 is total
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b are_maximal maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n[ -> nes] for \subseteq ;
not u in { ag } ;
the carrier of f c= B \/ C ;
reconsider z = x as Vector of V ;
cluster the carrier of L -> \rangle ;
r (#) H is empty implies r (#) H is convergent
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal algebra , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : x in : not x in { y } ;
let x , y be Element of X ;
let A , I be contradiction for f ;
[ y , z ] in O7 ;
( / Macro i ) . x = 1 ;
rng Sgm A = A & rng Sgm A = A ;
q |- \! \! \smallfrown All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z |^ Y ;
( DComput ( D2 , D1 , 1 ) ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , D :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative associative for non empty multMagma ;
x in support ( ( support t ) --> 1 ) ;
assume a in [: G ( ) , G ( ) :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 & b1 divides b2 ;
> upper_bound M1 & upper_bound M1 <= upper_bound M2 implies M1 - M2 <= M1 - M2
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= IC Comput ( P2 , s2 , k ) ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices H ;
( seq " ) is non-zero & ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hK1 c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 union X2 is SubSpace of X1 union X2
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster real-valued for Relation ;
ex v st C = v + W ;
let GX be non empty addLoopStr , x be Element of Y ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive antisymmetric RelStr , x be Element of L ;
R is reflexive transitive in X & R is transitive in X ;
E , g |= the_right_argument_of H implies E , g |= H
dom G `2 = a & dom G `2 = b ;
( 1 - 4 ) * ( - 1 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of F . i ;
D [ P-6 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of H ;
rng f\mathbb R c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = AM +* {} .= ( A \/ B ) \/ {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng g ;
M . ( F . 0 ) in REAL ;
holds holds holds holds ( Im a ) ^2 = ( Im b ) ^2
assume that the distance of V , Q and the distance of V , Q ;
let a be Element of op ( V ) ;
let s be Element of PH ( ) ;
let PA be non empty thesis , a be Element of Y ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= \hbox { ( j - 1 ) } ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in JumpParts ( JumpParts T ) & x in JumpParts T ;
let h be Morphism of c , a ;
Y c= [: R , R :] implies Y in [: R , R :]
A2 \/ A3 c= L1 \/ L2 \/ L2 ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x1 <> x2 ;
dom <* y *> = Seg 1 .= dom <* y *> ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> \hbox closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear implies q1 = q2 ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 - n ) / ( n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume that the carrier of L misses rng G and the carrier of L = {} ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT & dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= Arg a & Arg a < PI ;
o9 , a9 // o9 , y & o9 , b9 // o9 , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) /\ dom ( uncurry g ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT , x be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> natural for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ S c= conv @ A & conv @ S c= Int @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P ! ( P ! l ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y2 , T ;
W1 is_\HM { field W1 , W2 } implies R is well field R & R is well field R
assume x in the carrier of R & y in the carrier of R ;
dom nM = Seg n & dom nM = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in an implies [ a , y ] in the InternalRel of f ;
assume that I c= J and I c= K and I c= J ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin . x ) <> 0 & ( cos . x ) <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos * cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0
t3 . n = t3 . n .= s . n ;
dom ( ( - 1 ) (#) F ) c= dom F ;
W1 . x = W2 . x & W1 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( h . p2 ) `2 ;
GI = ( U /. 1 ) `1 .= U `1 ;
f . rp1 in rng f & f . rp1 in rng f ;
i + 1 + 1 - 1 <= len f - 1 ;
rng F = rng ( F . 0 ) .= rng ( F . 0 ) ;
mode `2 is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x1 . o in L2 . o ;
the carrier of \HM { m : m in B } c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
( seq ^\ k1 ) is lower & ( seq ^\ k ) is lower ;
len ( F | ( len F -' 1 ) ) = len F ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , x be Element of X ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be empty Element of be Element of of of of be ;
cluster empty -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j2 ;
redefine func J => y -> total for I -valued Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def6 : ( a - 1 ) / a = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o9 , b , b9 & LIN o9 , b9 , c9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FX2 be non empty thesis ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pA2 = x , pA2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict strict non empty <* M *> -> strict normal <* of M ;
rng c `1 misses rng ( e - a ) `1 ;
z is Element of gr { x } & z is Element of gr { y } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot - cot ) /\ dom ( cot - cot ) ;
the component of Q c= UBD A & UBD Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 + 1 ) /\ dom ( 1 + 1 ) ;
pred f = u means : Def6 : a * f = a * u ;
for n holds P1 [ \mathop { \rm VERUM } n ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
( n1 gcd n2 ) = 1 & ( n1 gcd n2 ) = 1 ;
set oI = a * ( 0. ( INT , p ) ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 and rng seq c= dom f ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n |^ 1 } \ { n } , h = n -tuples_on NAT , f = ( n -tuples_on NAT ) --> 0 , g = ( n -tuples_on NAT ) --> 1 ;
k = a or k = b or k = c ;
aa , ag , ag , ag , bg is_collinear ;
assume Y = { 1 } & s = <* 1 *> ;
Ik1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 .= y ;
cluster trivial -> finite for _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ \bullet implies A , B are_that B , A |^ n
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) / |. q .| ;
f1 is_\HM { the } \HM { \HM { the } \HM { non } \HM { empty } } ;
f `2 / |. q .| <= q `2 / |. q .| ;
h /. 1 in L~ Cage ( C , n ) ;
b `2 / |. b .| <= ( p `2 ) / |. p .| ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( f , g ) ) ;
p2 in NO & p2 in NO implies p2 in NO
len ( the_left_argument_of H ) < len ( H ) ;
F [ A , ( F . A ) . x ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def6 : A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 is_collinear implies A1 \/ A2 = A2 \/ A3
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C ( p , Sc ) & b in C ( p , Sc ) ;
then S is non empty and P-2 [ S ] ;
Cl Int [#] T = [#] T .= [#] T ;
f12 | A2 = f2 | A1 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 is conjunctive ;
1_ 1 c= ( \mathop { t } * ( ( p - 1 ) / ( p - 1 ) ) ) ;
0 * a = 0. R .= a * 0 .= 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v, vn = v4 /. n , vn = v5 /. n ;
r = 0. ( REAL-NS n ) & ||. x - x0 .|| < r ;
( f . p4 `1 ) ^2 >= 0 & ( f . p4 `2 ) ^2 >= 0 ;
len W = len ( W ) - len ( W ) .= len ( W ) - len ( W ) ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t16 / W8 does not destroy b1 & not t8 in dom b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 ] -> pair for set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be with_\hbox { NAT , { \mathbb N } } , A be non empty set ;
rng f = \ <* S , X *> ;
let p be Element of B , the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and Rx0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right & gR in the right of g ;
let A1 , A2 be Point of S , x be Point of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 implies 2 * 1 in Seg 3
reconsider X-5 = X , X\overline X = Y as non empty Subset of Tsuch
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 + len g2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y * i ) ;
( C * ( ( - 1 ) / p ) ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng ( D2 | Seg ( len D2 -' 1 ) ) ;
for p being Real st p in Z holds p >= a
( for x being Element of X holds f . x = proj1 . x ) implies f is continuous
( seq ^\ m ) . k <> 0 & ( seq ^\ k ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p } ) . 2 = d & ( p . 1 ) `2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) , g . ( mod P ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i , \hbox { \boldmath $ m $ } as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n & x in ( the Sorts of B ) . n ;
len that len that that that 1 in Seg len f2 and 1 <= len f2 ;
pA1 c= the topology of T & pA2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T2 ;
G * ( B * A ) = ( id o1 ) * ( id o2 ) ;
assume that p , u , q is_collinear and u , v , q is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , g = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , a1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 . O ) `2 ) ^2 <= 1 ^2 ;
p in LSeg ( E . i , F . i ) ;
II * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q7 . x in rng q7 & q7 . x in rng q7 ;
Carrier ( L7 ) misses Carrier ( L7 ) \/ Carrier ( L7 ) ;
consider c being element such that [ a , c ] in G ;
assume ( for o\in st o\in holds othesis ) & ( for o st o in I holds o <> holds o is thesis ) ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ CV ) " { 0 } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . ( j + 1 ) .] ;
pred 0 <= x & x <= 1 & x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 implies |. p .| * ( - q `2 ) = - q `2 * ( - q `2 )
redefine func \cal aS ( S , T ) -> Subset of S ;
let x be Element of [: S , T :] ;
the Arrows of F . ( a , b ) is one-to-one ;
|. i .| <= - ( 2 |^ n ) / ( 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
( } * ( n + 1 ) ! > 0 * <i> ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= A1 /\ A2 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & dom B <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G2 & y in X \/ { x } ;
set v2 = v4 /. ( i + 1 ) , v2 = v4 /. ( i + 1 ) ;
x = r . n .= seq1 . n .= ( seq1 + seq2 ) . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A1 :] ;
0 < ( p - ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . m3 ;
B \ominus X \/ B \ominus Y c= B \ominus X
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> being \mathop { \rm \hbox { - } o } -valued for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of Y ;
reconsider px0 = p . x , px0 = q . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and a in - X and a <= b ;
Int Cl A c= Cl Int Cl Int Cl A & Int Cl Int Cl A c= Cl Int Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 / |. p2 .| <= ( - 1 ) * ( - 1 ) ;
Cl ( Q ` ) = [#] ( TT ) ;
set S = the carrier of T , T = the carrier of S ;
set I8 = TOP-REAL ( n + 1 ) , I8 = TOP-REAL ( n + 2 ) ;
len p - n = len ( p - n ) .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nn} = nn} - nni as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s . f ) ;
let qbeing , q<* *> , q<* *> , q<* *> , q<* *> ;
a in the carrier of S1 & a in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) * S8 ) . x ;
consider x being element such that x in be S1 -such that x in B ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( o , h ) `1 , i1 = ( h , k ) `1 , i2 = ( h , k ) `1 , i2 = ( h , k ) `1 , i2 = ( h , k ) `1 , i2 = ( h
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 ;
let T1 , T2 be Scott topological or T2 is Scott being topological thesis of L ;
then x <= y & : x in : x in : x in { y } ;
set M = n -is non empty ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of a9 ) c= dom H & rng ( the_arity_of a9 ) c= dom H ;
z1 " = z9 " & z1 " = z2 " * z2 " ;
x0 - r / 2 in L /\ dom f & f /. x0 = r - r / 2 ;
then w is non empty & rng w /\ L <> {} ;
set x-10 = ( x ^ <* Z *> ) | ( X \ Z ) ;
len w1 in Seg len w1 & len w2 in Seg len w1 & len w2 in Seg len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of thesis , k be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. a .| ) ;
p `1 <= Gik `1 & p `1 <= G * ( len G , 1 ) `1 ;
rng ( g ) c= L~ ( g ) \/ LSeg ( g , 1 ) ;
reconsider k = i-1 * ( i + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } & F is \HM { -infty }
reconsider x9 = x9 , y9 = y9 as Vector of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y as Element of ( the carrier of X ) * ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag .= ( m + 1 ) * ( m + 1 ) ;
a / ( s . m - s . n ) / ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and C2 \/ C2 = C2 \/ C2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
- - 0. R = a & b-0 = b ;
FF is_closed_on t3 , Q7 & FF is_halting_on t3 , Q7 ;
set T = non such that for X , x0 holds X is non empty or T is non empty ;
Int Cl Int Cl R c= Int Cl R & Int Cl Int Cl R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . x } .= { F . x } ;
G-23 `1 \/ { c } c= B \/ S \/ S ;
f\rm is Relation between [: X , Y :] , X & X is Subset of [: Y , X :] ;
set RQ = the Point of P , RQ = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider pu = u , pv = v as Element of ( TOP-REAL n ) | ( Seg n ) ;
g . x in dom f & x in dom g implies g . x in dom ( f + g )
assume that 1 <= n and n + 1 <= len f1 and f1 /. n = f1 /. ( n + 1 ) ;
reconsider T = b * N as Element of ^ ( G , N ) ;
len P\bf 1 <= len P-35 & len P\bf 1 <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple function of S
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL m ;
rng f = the carrier of ( ( Carrier A ) \/ ( A . i ) ) ;
assume s1 = sqrt |[ 2 , p `2 / p `1 ]| ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of IQ ;
reconsider X0 = X , Y0 = Y as RealNormSpace , X = Y ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of t-3 , tt2 ;
Q [ e-14 \/ { vd } , f ] & not { vd } in f ;
g \circlearrowleft ( W-min L~ z ) = z implies ( g /. 1 ) .. z = ( g /. 1 ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle - v\vert v - y .| ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x ;
( 7 - p1 ) / ( 1 - e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan - cot ) `| Z ) . x in dom ( sec - cot ) ;
i2 = ( f /. len f ) `1 & i2 = ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) .= X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , v be VECTOR of V ;
dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
SFinSequence / ( g , f ) = ( S . g ) / ( g , f ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & ( Initialized p ) . 0 = p . 0 ;
i1 does not destroy i2 & i3 does not destroy b3 implies not ( i1 is not empty & not ( i2 is not empty & not i1 is not empty & not i2 is not empty )
arccos r + arccos r = ( PI / 2 ) * ( 1 + 0 ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & for x st x in Z holds f2 . x <> 0
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len f ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( not ( ex u st C in T ) & u <> TRUE ) implies not ( for x st x in T holds x in C )
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 in dom f ;
p2 `1 - x1 > - g / 2 - g / 2 - g / 2 ;
|. r1 - TOP-REAL n .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 & dom S-14 = Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .= D0W .3 + 1 ;
i1 = ma + n & i2 = ma + n & j2 = ca + n ;
f . a [= f . ( f . O1 , a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R /. 1 ) \/ LSeg ( R /. len R , 1 ) ;
set h = the continuous Function of X , R , a , b be Real ;
set A = { L . ( k . n ) : n in dom L } ;
for H st H is atomic holds P [ H ] ;
set b' = S5 ^\ ( i + 1 ) , Sa1 = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b ) ;
( 1 - s ) / ( n + 1 ) < ( 1 - s ) " ;
( l = [ dom l , cod l ] ) & ( l = [ cod l , cod l ] implies l = k ) ;
y +* ( i , y /. i ) in dom g & y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , x be Element of CQC-WFF ( Al ( ) ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
p2 in rng ( f /^ ( 1 + 1 ) ) & p2 in rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) + 1 <= len D2 ;
assume x in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( q } ) ) ) ) ) ) ) ) \/ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( q ) ) ) ) )
- 1 <= ( ( f2 . O ) `2 ) / ( 1 + ( - ( f2 . O ) `2 ) ) ^2 ) ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 - k2 .= k1 - k2 + k2 - k2 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( 1 , 1 ) .= 1 ;
consider u being Nat such that b = p |^ y * u ;
ex A being Line of B st a = Sum A & A is limit_ordinal of B ;
Cl ( union ( H ) ) = union ( ( Cl ( H ) ) /\ ( Cl ( H ) ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + len t2 ;
v-29 = v + w |-- ( A , v + A8 ) ;
cv <> DataLoc ( t3 . GBP , 3 ) .= DataLoc ( t3 . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= s ;
( \dot y ) . s = s . ( \dot y . s ) ;
{ s : s < t } in INT & t = {} implies t = {}
s ` \ s = s ` \ 0. X .= ( s ` \ s ) ` ;
defpred P [ Nat ] means B + $1 in A & not $1 in B + A ;
( 329 + 1 ) ! = 3329 ! * ( 329 + 1 ) ;
( U U ) . succ A = ( U U ) . ( ( U , A ) . ( ( U , A ) ) ) ;
reconsider y = y as Element of ( len y ) -tuples_on ( the carrier of K ) ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k , q = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) support z , g = ( S , U ) --> 0 ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , R^1 , p1 , p2 be Point of TOP-REAL n ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n + 1 ) -tuples_on the carrier of K , x be Element of K ;
reconsider l = 0. ( { 0. V } ) , r = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. x9 - gg .|| < r2 - g / 2 + g / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 implies b9 , c9 // c9 , a9
1 <= k2 -' k1 & k1 + 1 = k2 + 1 implies k2 = k2 + 1
( p `2 / |. p .| - sn ) / ( 1 + sn ) >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
( E-max C ) in cell ( RR , 1 , 1 ) /\ L~ R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `2 , a // b `1 , a `2 ;
g . n = a * Sum fk1 .= f . n * ( a * b ) ;
consider f being Subset of X such that e = f and f is empty ;
F | ( N2 , S ) = CircleMap * ( F | N2 ) .= ( F | N2 ) | N2 ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( x , r ) c= Ball ( x , r ) ;
the carrier of (0). V = { 0. V } .= { 0. V } .= { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( ( vseq . n ) - tseq . m ) - tseq . n .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = t22 as 0 string of S2 ;
reconsider x-29 = seq . n , xd = seq . n as sequence of REAL n ;
assume that E-max C meets L~ go and not E-max C in L~ pion1 and not W-min C in L~ pion1 and W-min C in L~ co ;
- ( Partial_Sums ( F ) . n ) < F . n - ( F . x ) ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d1 = dist ( x1 , z2 ) ;
2 |^ ( 2 |^ -' 1 ) = 2 |^ ( 2 |^ -' 1 ) - 1 ;
dom vG2 = Seg ( len dG2 + 1 ) .= dom vG2 ;
set x1 = - k2 + |. k2 .| , x2 = - k2 + 1 ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L1 + L2 ) ) c= I2 & the carrier of ( Carrier ( L1 + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \hbox { - } over {} ;
Z c= dom ( ( ( - 1 ) (#) sin ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r / 2 ;
^2 c= ConsecutiveSet2 ( A , *> , st L . 0 ) & L . 0 = A ;
E = dom Carrier ( L ) & Carrier ( L ) c= E & Sum ( L ) = Sum ( L ) ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= ( I . IC ss2 ) ;
pred x > 0 means : Def6 : ( 1 / x ) ^2 = x ^2 / ( 1 - x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) .= LSeg ( g , i ) ;
consider p being Point of T such that C = [: [. p , q .] , { p } :] ;
b , c are_connected & - C , - C are_connected implies a , b are_connected
assume f = id ( the carrier of OO ) & f = id ( the carrier of OO ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { v } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the carrier of G_ ( k , X ) & A2 in the carrier of G ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - i ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S\mathopen , j ) = S\mathopen ( j , i ) ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f - Re ( |. f .| * ( card b - a ) ) .| is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= 0 + 4 ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f| ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + y1 ;
fj . a = fj . a & v in InputVertices S & v in InputVertices S & v in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `1 <= ( E-max C ) `1 ;
consider p such that p = pSubset and s1 < p and p < s2 and p in LSeg ( s1 , i ) ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f /. 1 = f /. len f
dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } .= { {} } ;
consider r such that r _|_ a and r \not _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 in dom f ;
for L being complete LATTICE for A , B being Subset of L holds ( for x st x in O holds x in A ) implies A is isomorphic
[ gi , gj ] in II \ II ~ & [ gI , gj ] in II \ II ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom f1 /\ dom f2 holds f1 . r <> 0 ;
reconsider y = ( a ` ) ` , z = ( a ` ) ` as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) (#) f ) . c <= h . c ;
set G3 = the set of G , v = the Vertex of G , S = the set of G , S = the set of G , S = the carrier' of G , E = the carrier' of G , N = the carrier' of G , E = the carrier' of G , N = the carrier' of G , N
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / |. p .| < d / ( p |^ m ) ;
for x being element st x in for u being element st u in for t being element st t in : u <> t holds x in I holds x in I
P = the carrier of ( ( TOP-REAL n ) | P ) | Q .= ( TOP-REAL n ) | Q ;
assume p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom g ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , Y be Subset of Y , h be Function of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n implies m <= n ;
H * ( g " * a ) in the right of H * ( g " * a ) ;
x in dom ( ( cos / sin ) `| Z ) implies ( cos / cos ) . x = - cos . x / ( cos . x ) ^2
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p1 , P , p1 , p2 implies LE q2 , p2 , P , p1 , p2
attr B is an component of A means : Def6 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 & $2 = union rng $2 ;
n + - n < len ( pl + - n ) + ( - n ) ;
attr a <> 0. K means : Def6 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom thesis and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and not x1 in { x1 } ;
for n ex r being Element of REAL st X [ n , r ] & X [ n , r ]
set Ci1 = Comput ( P2 , s2 , i + 1 ) , Ci2 = P2 +* I ;
set cv = 3 / 2 -: ( a , b , c ) = 0 ;
conv @ W c= union ( F .: ( E " ( W /\ E ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 - |. v2 - v1 .| ) / ( 2 * ( 1 + ( 1 + ( 1 + 1 ) ) ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom f /\ dom f4 .= dom f /\ dom g ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq as Point of ( TOP-REAL n1 ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom *> <* *> & commute ( Frege ( A . o ) ) = ( Frege ( A . o ) ) . y ;
for I being non degenerated commutative Ring holds the carrier of I is commutative commutative doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* J ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( lp1 . i ) = ( v *' ( lp1 ) ) . i ;
consider n being element such that n in NAT and x = ( sn succ n ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x <> {} ;
Funcs ( X , 0 , x1 , x2 ) = { EE } & card { x1 , x2 } = k ;
j + ( 2 * k ) + m1 > j + ( 2 * k ) + m1 ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3
mg1 . HT ( mg2 , T ) = 0. L .= 0. L ;
then H1 , H2 are_<* H1 , H2 *> & card H1 , card H2 are_\kern1pt ;
( ( N-min L~ f ) .. f ) .. f > 1 & ( ( N-min L~ f ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( ( TOP-REAL 2 ) | L~ g ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of T ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I V V V V V V V \mathop \mathop { d } = ( k2 + 1 ) - 1 .= k2 - 1 ;
[: u , { u9 } :] = { [ a , u9 ] } .= { [ a , b ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W2 ;
for y st y in rng F ex n st y = a |^ n & a |^ n = b |^ n
dom ( ( g * ( f . ( V \dot \to C ) ) ) | K ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) . s ) & x in ( ( the Sorts of U0 ) . s ) ;
ex x being element st x in ( and ( Carrier ( O) ) . s ) & x in ( Carrier ( O) ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p00 } /\ LSeg ( p1 , p2 ) ;
( b + ( bs0 - b ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A and z <> y ;
( the sequence of ( ( the carrier of X ) --> 0 ) ) . x <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q `2 = ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` .= g | E-4 ` ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , j1 = x4 as Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f |^ n0 is L & f . n0 is L ;
Seg len ( ( f1 ^ f2 ) | ( Seg len ( f2 ^ g2 ) ) ) = dom ( ( f1 ^ f2 ) | ( Seg len ( f2 ^ g2 ) ) ) ;
( Complement ( A . m ) ) . n c= ( Complement ( A . n ) ) . m ;
f1 . p = p9 & g1 . p = d & g1 . p = b & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= F | Y ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| |^ n ) / ( n + 1 ) <= ( r2 |^ n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for i st i in dom F holds ( F . i ) . i = g . i ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 and W2 is Subspace of W3 ;
||. t-15 . x .|| = lim ||. ( x - y ) .|| .= ||. ( x - y ) .|| .= ||. x .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) - ( - ( p `2 ) ) ) / ( 1 + ( - ( p `1 / |. p .| - cn ) ) ) ^2 <= ( - ( - ( p `2 / |. p .| - cn ) ) ) / ( 1 + ( - ( p `2 / |. p .| - cn ) ) ) ^2 ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( Sphere ( p , r ) ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B `1 .= B * ( i , j ) ;
attr a <> 0 means : Def6 : ( A \+\ B ) Let a = ( A Y. a ) \+\ ( B f2 ) ;
then f is_\mathbin { is_} pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 and a > 0 ;
w1 , w2 in Lin { w1 , w2 } implies w1 = w2 or w2 = w1 or w2 = w2
p2 /. IC s-7 = p2 . IC sU .= ( 0 + 1 ) - 1 .= ( 0 + 1 ) - 1 .= 0 ;
ind ( T-10 | b ) = ind b .= ind B - ind ( T-10 | b ) .= ind B - ind ( T-10 | b ) ;
[ a , A ] in the carrier of \hbox { - } or [ a , A ] in the carrier of \hbox { - } as Subset of TOP-REAL 2 ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o2 ) ;
( ( a , CompF ( PA , G ) ) ) . z = FALSE & ( a , CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = I /. ( 11 + 1 ) as Element of ( S , U ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in the carrier' of [: A , B :] & [ f21 , f22 ] in the carrier' of [: A , B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( (0). V1 ) .= { 0. V1 } .= { 0. V1 } ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 . 1 = P2 . len P2 ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. |[ b , c ]| = c .= c /. |[ a , c ]| .= c /. |[ a , c ]| ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 1 - r ) * ( 1 - r ) * ( 1 - r ) in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= ( h . p2 ) `2 + D .= ( h . p2 ) `2 ;
R . b - b - a = 2 * - b .= 2 * b - b .= b - a ;
consider \hbox such that B = - 1 * ] + ( 1 - \hbox { - } 1 } * A ) and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , I ) ) .= dom ( a * ( the Sorts of A ) ) ;
[ P . ( U6 ) , P . ( l6 ) ] in => ( ( 'not' P ) . ( l6 ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of ( TOP-REAL 2 ) | ( L~ z ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the right of g & x = g . x ;
consider M being strict Subspace of Aj such that a = M and T is strict Subspace of M ;
for x st x in Z holds ( ( #Z 2 ) * f + #Z 2 ) . x <> 0 & ( #Z 2 ) * f + #Z 2 = f . x
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + 1 + m .= len W3 + 1 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian Lipschitzian Lipschitzian of X , Y ;
( i - 1 mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is <* s1 , s2 *> and F in the O of s2 and F in the O of s1 and F in the O of s2 ;
( ( ( ( ex x , y st x in ( p - q ) ) ) / ( p - q ) ) ) * ( ( p - q ) / ( p - q ) ) = ( p - q ) / ( p - q ) ;
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u
for B be Subset of u-5 st B in E holds A = B or A misses B or B = C ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 \/ W2 , W3 = p \/ W2 ;
x in { X where X is Ideal of L |^ \rm op ( L ) : X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b holds a + b * id a = in Hom ( a + b , a + b ) ) implies a = b
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( i - 2 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & b on P & c , d on P & a , b // P & c , d // P ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in V ;
n in { i where i is Nat : i < n0 + 1 & i < n0 + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn * |. p .| & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) .: ( A , O1 ) .= A . ( succ O1 ) ;
set Ii1 = Macro AddTo ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 2 ) , Ii2 = goto ( card I + 2 )
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. ( i + 1 ) <> z /. ( i + 1 )
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & X c= the carrier of L1 implies L1 is strict Subspace of L2
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and P [ x9 , x , x9 ] ;
reconsider ee = e4 , fe = f-5 , fe = f-5 as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. <= ( card ( the carrier of V ) ) * ( M . O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and x in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A in succ ( A + $1 ) ;
the left of - g = the left of g & the left of - g = the left of g implies g = f ;
reconsider p\mathopen = x , pj = y , pj = z as Point of ( TOP-REAL 2 ) | K1 ;
consider g3 such that g3 = y and x <= g3 and g3 <= x0 and for x st x in dom f holds f . x = f . x ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len y2 .= len ( x2 ^ y2 ) + len y2 ;
for x being element st x in X holds x in the set of set & x in the set of set & y = ( the Element of n0 ) . x ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func such of X -> set means : Def6 : for x being set holds x in it iff x is real hh\pi & it = id X ;
len ( { f /. ( len Cf ) , Cf /. 1 ) } ) <= len Cf + len Cf - 1 ;
attr K is with_a , v means : Def6 : a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is \rm - 1 / f . x
IC Comput ( P-6 , smeans : Def6 : for k st k in dom Pd holds Pd . k = ( d . k ) * ( d . k ) ) & ( d . k ) * ( d . k ) = ( d . k ) * ( d . k ) ;
pred q < s & r < s & ]. r , s .[ \not c= ]. p , q .[ implies p in ]. r , s .[
consider c being Element of Class f such that Y = ( F . c ) `1 and ex x st x in X & c in Y ;
func the ResultSort of S2 -> Function means : - the ResultSort of S1 = id the carrier' of S2 & for x being set st x in the carrier' of S2 holds it . x = ( the ResultSort of S2 ) . x ;
set yp1 = [ <* y , z *> , f2 ] , yp2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) * ( arccot ) ) `| Z ) . x ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f \/ L~ f \/ L~ f \/ L~ f \/ L~ f \/ { f /. ( len GoB f ) } , ( L~ f ) \ L~ f \/ L~ f ) ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - len f + 1 - 1 + 1 + 1 - 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + 1
for n ex x st x in N & x in N1 & h . n = - ( x0 - r ) / ( n + 1 )
set sx0 = ( > 0 , a , I ) . i , sx0 = ( a , I ) . i , sx0 = ( a , I ) . i , sx0 = ( a , I ) . i , sx0 = ( a , I ) . i , sx0 = ( a , I ) . i , sx0 = ( a , I ) . i , sx0
p ( ) . k = 1 or p ( ) . 0 = - 1 or p ( ) . 0 = 1 or p ( ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and not x in V1 and x in V1 and y in V1 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p - len p ) ;
g + h = gg + hg1 & A1 + h = gg + h & A2 + h = gg + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f \leftarrow x ) implies f / x = f / y & f / y = f / x ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , M ) = rpoly ( 1 , t ) *' *' t + 0. F_Complex .= 1 + t *' t .= 1 + t *' t ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = X
( ( N-min X ) `1 ) ^2 <= ( ( ( N-min X ) `1 ) ) ^2 & ( ( ( N-min X ) `2 ) ) ^2 <= ( ( ( NW-corner X ) `1 ) ) ^2 ;
for c being Element of the \geq the \geq A , a being Element of the free of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b = 0 or a = 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , m , n ;
set x2 = |( Re y , Im ( x - Im y ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n - 0 .| < ( e / 2 ) * ( ( n + 1 ) + 1 ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / ( 2 * ( b-a - 1 ) ) ) / ( 2 * ( b-a - 1 ) ) ;
for x , y being set st x in R" holds x , y are_\hbox { F . x }
deffunc F ( Nat ) = b . $1 * ( M * G ) . $1 * ( M * G ) . $1 ;
for s being element holds s in -> ( f 'or' g ) iff s in -> Element of \rm \rm \rm : _ ( f )
for S being non empty non void non void holds S is with_holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `1 ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = np1 '&' ( M . x qua Element of BOOLEAN ) , nw = M . ( n + 1 ) , nw = M . ( n + 1 ) , nw = M . ( n + 1 ) ;
f " V in ' ( X ) & f " V in D & f " V in D & f . ( the carrier of X ) = ( the carrier of X ) \ { p } ;
rng ( ( a ^\ c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being such that y `1 = y and dom y `1 = WWWWthesis and y `2 = WWWW
dom ( 1 / f ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 .[ /\ ]. x0 , x0 + r .[ ;
non empty Subset of non empty Subset of non 1 , r , - r , - r , - r , - r be Element of TOP-REAL n ;
v ^ ( n-3 |-> 0 ) in Lin ( rng ( B-9 | c1 ) ) & v ^ ( n-9 | c1 ) = v ^ ( n-9 | c1 ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i = k2 := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( 5 , succ i1 ) .= succ ( 5 , succ i1 ) .= ( NAT --> succ i1 ) . NAT .= ( NAT --> succ i1 ) . NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & not LIN a , a9 , c & not LIN a , a9 , c
( L1 x x ) := O c= ( L1 => O ) => ( L2 => O ) .= ( L1 => L2 ) => O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( u1 - w ) = b * ( y - w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| implies Sum ( $1 ) = Sum ( $1 ) ;
u = cos / sin . ( x , y ) * x + ( cos / cos . ( x , y ) * y ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ <* p *> , {} _ { A } ] implies P [ p ]
consider X being Subset of CQC-WFF ( Al ( ) ) such that X c= Y and X is finite and X is inininand X is ininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) ) .. Cage ( C , n ) - 1 ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h . l1 & g <= f . l1 & l1 <= h . l1 } ;
Ser ( ( G . n ) vol ) <= ( Partial_Sums ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( . . n . . n ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . n ;
f . y = x .= x * ( 1_ L ) .= x * power L .= x * power L .= x * ( y , 0 ) ;
NIC ( <% i1 , succ i1 %> , k ) = { i1 , succ i1 } .= { succ i1 , succ i1 } .= { succ i1 , succ i1 } .= { succ i1 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( Carrier ( ( Carrier ( I ) ) ) +* ( i , { 1 } ) ) ) in Z . ( i + 1 ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s2 , n ) .= Following ( s2 , n ) ;
W-bound Qs2 <= q1 `1 & ( for i st i in dom Qs2 holds ( i <= len Qs2 ) & ( i <= len Qs2 ) implies ( i <= len Qs2 ) & ( i <= len Qs2 implies ( i <= len Qs2 ) ) & ( i <= len Qs2 implies ( i <= len Qs2 ) + 1 ) = ( i + 1 ) + 1 ) ;
f /. i2 <> f /. ( len f -' 1 + len g -' 1 , f /. ( len f -' 1 ) ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) |= H / ( x. 4 , a ) ;
len ( ( P ^ ) . ( len ( P ^ ) ) ) in dom ( ( P ^ ) . ( len ( P ^ ) ) ) ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| < a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = f . ( n1 + 1 ) ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) - ( id the carrier of V ) .| ) implies v = 0. V
for phi holds phi in X implies not ( phi in X & not phi in X & phi in X ) & ( not phi in X & phi in X implies phi in X )
rng ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f |
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & a = c . 1 ;
( the_arity_of F , b , c ) = <* F , F , c *> & ( for a , b holds o <> a & o <> b implies o = a ) implies o = c
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = f . 1 ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & a3 = b1 & a4 = b2 & a4 = b3 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) .= D1 . n1 ;
f . ( ||. r .|| ) = ||. |[ r .|| , r ]| .|| /. 1 .= <* r *> . 1 .= r .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and Cis convergent and Cis convergent and lim Cis convergent and lim Cis convergent and lim Cseq = lim Cseq ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def6 : for b being Element of X holds F -Sum ( { b } _ f ) = f . b ;
p = - ( - ( p0 `1 / |. p0 .| - cn ) ) * ( ( - ( cn - cn ) ) / ( 1 - cn ) ) .= 1 - ( ( - ( cn - cn ) ) / ( 1 - cn ) ) * ( ( - ( cn - cn ) ) / ( 1 - cn ) ) .= ( - ( cn - cn ) ) * ( ( - ( cn - cn ) ) / ( 1 - cn
consider z1 such that b , x3 , x1 is_collinear and o , x1 , x1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear ;
consider i such that Arg ( Rotate ( f , p ) ) = s + Arg q + ( 2 * PI * i ) and Arg ( Rotate ( f , p ) ) = 2 * PI * i ;
consider g such that g is one-to-one and dom g = card f and rng g = f . x and g . x = f . y ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and P2 <> {} and Q2 <> {} and P2 <> {} ;
attr F is associative means : Def6 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x `2 < i or m in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and ( for k st k in dom P-2 holds P`1 . k = ( P`1 . k ) `1 ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & ( for n holds seq . n = r * seq . n ) & ( for n holds seq . n = r ) implies seq is convergent & lim seq = r
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , [ a , b ] ] .= [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p "\/" q = p "\/" q } .= { p "\/" q where q is Element of L : q in D2 } ;
consider z being element such that z in dom ( ( F . z ) | ( dom F ) ) and ( ( F . z ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , b1 , b2 ) ) . ( \mathbb j ) .= ( Mx2Tran J ) . ( \mathbb j ) ;
- 1 / ( - 1 ) * D = mm1 (#) D | n .= mm1 (#) D .= mm1 (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) .= ( - 1 ) (#) ( - 1 ) ;
attr for x be set st x in dom f /\ dom g holds g . x <= f . x & - g is nonnegative ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) \/ RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k \ a |^ k ;
k -th -ininininininininininininininin-inin-inin-in-in-inin-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in-in
for s being State of Aj holds Following ( s , n . 0 + ( n + 2 ) * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is continuous
support ( support ( n ) ) \/ support ( ( support ( m ) ) ) c= support ( ( support ( n ) ) ) \/ support ( ( support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) --> the carrier of C , the carrier of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi ( succ b1 ) . a = g . a & phi ( b ) . a = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i + j in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 } \/ { x3 , x4 } .= { x1 } \/ { x2 , x3 , x4 } .= { x1 , x2 , x3 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies U1 = U2
( - ( 2 * a * ( b - a ) ) / ( 2 * a ) ) ^2 - delta ( a , b , c ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] and ( for z being element holds z in N ~ N implies z in N ) ;
assume ( the Arity of S ) . o = <* a *> & ( the ResultSort of S ) . o = r & ( the ResultSort of S ) . o = <* r *> & ( the ResultSort of S ) . o = r ;
Z = dom ( ( #Z ( n ) * ( arccot - arccot ) ) / ( f1 + #Z 2 ) ) ) /\ dom ( ( #Z 2 ) * ( arccot - arccot ) ) ;
integral ( f , SS1 ) is convergent & lim ( \HM { the carrier of S , the carrier of T :] ) = integral ( f , SS1 ) ;
( for a holds ( a . ( f . a ) => ( g . a ) ) ) => ( x9 => ( g . a ) ) in
len ( M2 * M3 ) = n & width ( M3 ~ * M2 ) = n & width ( M1 * M3 ) = n & len ( M2 * M3 ) = n ;
attr X1 \/ X2 is open SubSpace of X means : Def6 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated ;
for L being upper-bounded antisymmetric RelStr for X being non empty RelStr for X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1= F1 . ( b `2 ) , f-1= F2 . ( b `2 ) , f-1= F2 . ( b `2 ) as Function of M . b , M . b ;
consider w being FinSequence of I such that the InitS of M , w -\geq <* s *> ^ w ^ w ^ w , q ^ w ^ w ^ w ^ w ^ w ^ w , q ^ w ^ w ^ w *> ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= ( g . a ) |^ 0 ;
assume for i be Nat st i in dom f ex z be Element of L st f . i = rpoly ( 1 , z ) & z in D & f /. i = z * z ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} & K is closed & L is closed ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o-21 = o `1 , op = o `2 , op = o `1 , op = o `2 , op = o `1 as Element of TS ( ( the Sorts of A ) . o ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , \dots , 0 } , 0 } .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) ;
Ez " . 1 = ( Ez qua Function ) " . 1 .= ( ( 1 - 2 ) * ( 1 - 2 ) ) " . 1 .= ( 1 - 2 ) * ( 1 - 2 ) .= ( 1 - 2 ) * ( 1 - 2 ) ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 - |. M .| ) * ( M * ( 1 , 1 ) ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , ( ( Gauge ( C , n ) * ( i + 1 , 1 ) ) ) ) /. ( i + 1 ) , ( ( Gauge ( C , n ) * ( i + 1 , 1 ) ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x ) + R /. ( x- x ) ;
g . c * ( - g . c * f . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) ;
assume that ColVec2Mx f in the set of \HM { the } \HM { set } , ColVec2Mx b = ( ColVec2Mx b ) * ( ColVec2Mx b ) and len f = width A and width A = width B and len B = len A and width A = width B and width A = width B ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( TOP-REAL n ) | ( the carrier of n ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 implies pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( b - a ) & Arg ( - a ) = Arg ( b - a ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the REAL of a , b ) & not c in Intersection ( the REAL of a , b )
assume that V1 is linearly-independent and V2 is linearly-independent and V1 = { v + u : v in V1 & u in V1 & v in V2 } and V1 = V1 \/ V2 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in N
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg card ( ( dom ( Pk1 ) " * Sk1 ) ) .= Seg card ( dom ( Pk1 ) ) .= Seg card ( dom ( Pk1 ) ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b and b <= s2 . n ;
h2 " . n = h2 . n " & 0 < - ( 1 / ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ GV ) * v & - w = ( - 1_ GV ) * w & - w = ( - 1_ GV ) * w & - w = ( - 1_ GV ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b be non zero Nat st a , b are_relative_prime holds ( for n be Nat holds support ( a * b ) = support ( a ) + support ( b ) ) & ( for n be Nat holds n in dom a implies n <= len b ) & n <= len b implies n = 1
consider A9 being countable set such that r is Element of CQC-WFF ( Al ( ) ) and A9 is ( len A ) -V ( ) ) `1 & ( for n being Nat holds A . n = ( A ( ) ) `1 ) & ( A ( ) ) `2 = ( A ( ) ) `2 ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { y2 } :] \/ [: { x2 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Lower_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime means : such : for p being prime Nat holds it is prime & for p being prime Nat holds p divides m & p divides n implies p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G & e . 7 in G ;
( r (#) o ) . ( 2 * n ) . x = ( r (#) delta ( h ) ) . ( 2 * n + ( n + 1 ) ) . x ;
j + 1 = ( len h11 + 1 ) - 1 + 1 .= i + 1 - 1 + 2 - 1 .= i + 1 - 1 + 2 - 1 .= i + 1 - 1 + 2 .= i + 1 - 1 + 2 ;
( *' ( S , T ) ) . f = *' ( S , T ) . ( f , g ) .= S . ( ( *' ( S , T ) ) . f ) .= S . ( ( *' ( S , T ) ) . g ) .= S . ( ( *' ( S , T ) ) . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L1 ) and Sum ( L1 ) = Sum ( L2 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is *> means : Def6 : for p , q st p in R & q <> q holds ex P st P is special arc & p in P & q in P & q in P ;
dom product ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) ;
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ \mathop { \rm UpperArc } ( C ) ) ) <= upper_bound ( proj2 .: ( C /\ \mathop { \rm UpperArc } ( C ) ) ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - S . ( m - p ) .| < r
i * fN - fN = i * fN - ( i * fN - ( i * yN ) ) .= i * ( fN - ( i * fN ) ) - i * fN .= i * ( fN - ( i * fN ) ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) and for x being set st x in 2 -tuples_on Y holds f . x = F ( x ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C and g2 in C ;
func d |-count n -> Nat means : such : d |^ n divides d |^ n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `2 ) / |. q .| ) ^2 + ( ( q `2 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 - len h11 + 2 ) .= h21 . ( i + 1 - len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & a >= b & b >= a
||. h1 .|| . n = ||. h1 . n .|| .= ||. h .|| . n .= ||. h .|| . n .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( ( - ( #Z n ) ) * ( #Z n ) ) . x = f . x - ( #Z n ) . ( ( #Z n ) * ( #Z n ) ) .= ( - 1 ) * ( #Z n ) .= ( - 1 ) * ( #Z n ) ;
pred r = F .: ( p , q ) means : Def6 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = F . i ;
( r\mathbin { / 2 ) ^2 + ( r\mathbin { / 2 } - ( r\mathbin { / 2 } - ( r\mathbin { / 2 } - ( r\mathbin { / 2 } - ( r\mathbin { / 2 } - ( r\mathbin { - 1 ) ) / 2 ) ) ) / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( L * M ) @ ) & Det M = Sum ( L * M )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j - 1 ) * ( q *' ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p ) * r3 .= ( p . j - 1 ) * r3 .= ( p . j - 1 ) * r3 ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 - ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 ) ;
assume that the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H .= n + 1 -H ;
( ( O = 0 ) & ( O = 0 & O = 1 & O = 1 & O = 1 ) & ( O = 1 implies O = 1 ) & O = 1 implies O = 1 ) & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( 1 + 1 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a = ( - e ) / d implies ( a = ( - e ) / d ) & ( a = ( - e ) / d ) & ( - b ) / d = ( - ( - e ) ) / d ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= D ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * a * v & u in I & v in I & v in I
g `1 * P `2 * g `2 " = g `2 * ( g `1 * P `2 ) * g `2 .= g `2 * ( g `2 * P `1 ) .= g `2 * ( g `2 * P `1 ) .= g `2 * ( g `2 * P `1 ) ;
consider i , s1 such that f . i = s1 and not ( ex s st s = s1 & not ( ex s st s = s1 & s is empty ) & not ( ex s st s is empty & s is empty ) & not ( ex s , s1 st s is empty & s1 is empty ) ) ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s3 , t2 ] are_connected & [ s3 , t2 ] , [ s3 , t2 ] are_connected ;
then H is negative & H is not negative & H is not conjunctive & H is not conjunctive -g\mathopen \neq H & H is not negative -gex f st f is not negative -gex g st g is not negative -gex f st f = g & f is not conjunctive -gex g st g is not contradiction ;
attr f1 is total means : Def6 : 1 / 2 is total & ( for c st c in dom f1 holds f1 . c = c * ( f2 . c ) ) & ( for c st c in dom f1 holds f1 . c = c * ( f2 . c ) ) implies f1 is total & f2 is total ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2
p = 1 * p .= a " * a * p * q .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= ( a " * b " * b * q ) * ( b * q ) .= ( a " * b " * b * q ) * ( b * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ k ) <= upper_bound ( ( seq1 ^\ k ) ^\ ( n + 1 ) )
E-max C meets L~ go \/ L~ pion1 or E-max C meets L~ pion1 or E-max C meets L~ pion1 or E-max C meets L~ pion1 or E-max C meets L~ pion1 or E-max C meets L~ pion1 or E-max C meets L~ co or E-max C meets L~ co or E-max C meets L~ co or E-max C in L~ co & E-max L~ godo in L~ co ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k - K to_power k ) ;
assume h = ( ( B .--> B ' +* ( C .--> D ) ) +* ( E .--> F ' ) +* ( F .--> J ' ) ) +* ( J .--> F ' ) +* ( M .--> A ' ) +* ( N .--> A ' ) +* ( N .--> A ' ) +* ( N .--> A ' ) +* ( N .--> A ' ) ) ;
|. ( ( upper_volume ( H . n , T ) || A ) . k - ( ( upper_volume ( H1 , T ) || A ) . k ) ) .| <= e * ( ( \HM { the } \HM { function } ) || A ) ;
( ( ( the Sorts of A ) . i ) ) . e = [ the \rbrace at v , the carrier of ( ( the Sorts of A ) . i ) --> ( the carrier of ( ( the Sorts of A ) . i ) ) ] ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 } } = { x1 , x1 } \/ { x1 , x2 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * cos ) = 0 and integral ( ( #Z n ) * cos ) = 0 and for x st x in A holds ( ( #Z n ) * cos ) . x = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( Sgm Y ) " * p & p `2 " * Sgm X = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p ;
for x , y st x in A holds |. ( 1 / ( f . x ) - 1 / ( f . y ) ) * ( f . y - 1 / ( f . y ) ) .| <= 1 * |. f . x - 1 / ( f . y ) .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) - sn / ( 1 + sn ) ) .= ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of RNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f holds ( for x be Element of NAT st x in dom f holds f /. x is compact ) & ( for x be Element of NAT st x in dom f holds f /. x is compact ) implies f /. x is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A ) ) . x = TRUE ) ;
consider FF such that dom FF = n1 and for k be Nat st k in n1 holds Q [ k , FF . k ] and for k st k in n1 holds Q [ k , FF . k ] holds F ( k ) = F ( k ) ;
ex u , u1 st u <> u1 & u , u1 / ( a , b ) / ( a , v1 ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) = u1 / ( a , b ) / ( a , b )
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` A ) = N ` A * N ` B
for s be Real st s in dom F holds F . s = integral ( R / ( R to_power 0 ) ) - integral ( ( f + g ) / ( f + g ) - ( f + g ) / ( f - g ) ) . x )
width AutMt ( f1 , b1 , b2 ) = len b2 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b2 .= len b2 .= len b1 + len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , PI / 2 .[ & for x st x in ]. - 1 , PI / 2 .[ holds f . x = - 1 / 2 * x + 1 / 2 * x + 1 / 2 * x * x ;
assume that X is closed w.r.t. being set and a in X and a c= X and y in a ^ { [ n , x ] } \/ y and x in a \/ y and y in a ;
Z = dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( arctan + arccot ) ) .= dom ( ( #Z 2 ) * ( arctan + arccot ) ) /\ dom ( ( #Z 2 ) * ( arctan + arccot ) ) ;
func > V . l -> Subset of V means : Def6 : for k st 1 <= k & k <= len l holds it . k in V & not it . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N st c is_be cluster cluster for net of N for net of L st c is cluster cluster strict for net of N holds c is convergent
for s being Element of NAT holds ( for v being Element of NAT holds ( for x being Element of C\mathop ( CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= 1 + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and for x st x in Z holds f . x = x / ( sin . x ) ^2 and f . x > 0 ;
for R being add-associative right_zeroed right_complementable left distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + y2 ) .= Seg len ( x (#) z ) .= dom ( x (#) z ) .= dom ( x (#) z ) .= dom ( x (#) z ) .= dom ( x (#) y ) ;
for S being Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . c ) & ( Obj S ) . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & \rrangle in \rrangle & $ \rrangle in \mathop { f . a , f . a } & { f . a } = { f . a } ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' H2 / ( x. 4 , x. 0 ) '&' H2 / ( x. 4 , x. 0 ) ) ;
for C1 , C2 being the carrier of C1 , f , g being stable Function of C1 , C2 st `1 = `1 & `1 = C2 holds f = g & g = h & f = g implies f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ ( W-bound L~ co ) / ( 2 |^ ( 1 + 1 ) ) .= W-bound L~ go \/ E-bound L~ co .= W-bound L~ co \/ E-bound L~ co L~ co .= W-bound L~ co ;
assume that u = <* x0 , y0 , z0 *> and f is_is_is_is_or SVF1 or SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `1 = x & ( t . {} ) `1 = x & ( t . {} ) `1 = x & ( t . {} ) `1 = y & ( t . {} ) `1 = y & ( t . {} ) `1 = y & ( t . {} ) `1 = y ) ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T ~ st a = f . x & b = f . y holds a >= b & b >= y ;
func Class R -> Subset-Family of R means : Def6 : for A being Subset of R holds A in it iff ex a being Element of R st a = Class ( R , a ) & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( \HM { the } \HM { vertices } ) . $1 ) `1 ) `1 ) `1 c= G . ( ( the non empty set ) `1 ) & ( ( ( the D of G ) . $1 ) `2 ) `1 c= G . ( ( the carrier' of G ) . $1 ) `1 ) ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and ( dim ( W2 ) = 0 implies ( dim ( W1 ) = 0 & dim ( W2 ) = 0 & dim ( W1 ) = 0 & dim ( W2 ) = 0 ) & ( dim ( W1 ) = 0 implies ( dim ( W1 ) = 0 & dim ( W1 ) = 0 ) ) ;
mamas ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= [ m . t , the carrier of C ] `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= d22 ;
consider g such that x = g and dom g = dom fx0 and for x be element st x in dom fx0 holds g . x in fx0 & g . x = ( f | X ) . x ;
x + 0. F_Complex |^ len x = x + len x |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x `1 + ( x + len x ) .= x `2 + 0. F_Complex .= x `1 + 0. F_Complex .= x `1 + 0. F_Complex .= x `2 + 0. F_Complex .= x `1 + 0. F_Complex .= x `2 + x `2 ;
( k -' ( k + 1 ) ) in dom ( f /. ( k -' 1 ) ) & ( f /. ( k -' 1 ) ) = ( f /. ( k -' 1 ) ) * ( f /. ( k + 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 \/ P2 \/ P2 and P1 = P1 \/ P2 \/ P2 \/ P2 \/ P2 \/ P1 \/ P2 \/ P2 \/ P1 \/ P2 \/ P2 \/ P2 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 \/ P2 \/ P2 \/ P2 and P1 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2
reconsider a1 = a , b1 = b , b1 = c `1 , c1 = p `1 , c2 = p `2 , c2 = p `2 , c2 = p `1 , c1 = p `1 , c2 = p `2 , c2 = p `2 , c2 = p `1 , c1 = p `2 , c2 = p `2 , c2 = p `2 , _ = p `1 , _ = p `1 , c1 = p `1 , c2 = p `1 , c2 = p `2 , _ = p `1 , a1 = p `1 , c1 = p `1 , c1 = p `1 , c2 = p `1 , c2 = p `1 , c2 = p `2 , c2 = p `1 , c2 = p `1 , c2 = p `1 , c1 = p `2 , c2 = p `1 , c2 = p `1 ,
reconsider set set set set set set set set set set set = G1 . ( t , b ) * F1 . f , F2 = G1 . ( t , a ) * F2 . f , F2 = G2 . ( t , a ) * F2 . f as Morphism of G1 , G2 . ( t , a ) * F2 . a ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n -P . m ) + Integral ( M , P . m -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & f1 . ( x , y ) = f2 . ( x , y ) holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 ) - ( G * ( i + 1 , 1 ) `1 ) ) / 2 , ( G * ( i + 1 , 1 ) `2 - ( G * ( i + 1 , 1 ) `2 ) ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer , b being Integer st a |^ i = b |^ i holds a |^ i = b |^ i * a |^ ( b |^ i )
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p where p is Point of TOP-REAL 2 : P [ p ] & p `2 <= 0 & p <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
( ( ( N-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-bound C ) / 2 ) ) / 2 <= ( ( N-bound C - ( S-bound C - ( S-bound C - S-bound C ) / 2 ) ) / 2 ) * ( ( S-bound C - ( S-bound C - S-bound C ) / 2 ) ) / 2 ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| <= P . x
len ( @ ( @ p ^ @ q ) ) = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len ( @ q ^ @ p ) .= len ( @ q ^ @ p ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) = m3 / ( x. 0 , m2 ) / ( x. 0 , m2 ) ;
consider r be Element of M such that M , v2 / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) |= r / ( x. 0 , m3 ) ;
func w1 \ w2 -> Element of Union ( G , R|. ) means : for i , j st i in dom ( ( ( ( ( G , i ) --> ( G . j ) ) ) | ( ( G . i ) --> ( G . j ) ) ) ) holds it . i = ( ( ( G , i ) --> ( G . j ) ) | ( ( G . j ) --> ( G . j ) ) ) . i ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . ( n + k ) + ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums ( seq ) ) . ( n + k )
set F = S \! \mathop { {} } ;
( Partial_Sums ( seq ) . K + Sum ( seq ) ) . ( K + 1 ) >= ( Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . ( K + 1 ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( 1 - x ) ) + R . ( x - x0 ) ;
func \HM { a , b , c , d \HM { a } , c , d } -> Subset of TOP-REAL 2 equals ( the +* ( a , b , c ) ) \/ ( the +* ( a , b , d ) ) ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( b * c ^2 + c * a ^2 ) + ( c * a ^2 + b * c ^2 + c * a ^2 ) >= 6 * a * b * c * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
+ ( Q ^ <* x *> , M1 ) = ( + ( Q , M1 ) +* ( L , { x } ) ) +* ( card ( { x } --> FALSE ) , ( L , x ) +* ( L , { x } --> FALSE ) ) .= ( M , ( M , { x } --> TRUE ) ) +* ( L , x ) ;
Sum ( F ) = r |^ ( n1 + 1 ) * Sum ( Cz ) .= C ( n1 ) * ( n + 1 ) .= Cz . ( n + 1 ) * ( Cz ) .= ( Cz ) . ( n + 1 ) * ( Cz ) .= ( Cz ) . ( n + 1 ) * ( ( n + 1 ) + 1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) . $1 ) . ( n + 1 ) = ( a * ( ( $1 + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) * ( ( n + 1 ) * ( n + 1 ) ) ;
( the_arity_of g ) . g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X , Y ) to_power Z tolerates X to_power Z & card ( ( X , Y ) to_power Z ) = card ( X Z ) & card ( X , Y ) = card ( Y ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . ( n + 1 ) holds b = N . ( s . n ) \ G . s ;
E , f |= All ( x. 2 , All ( x. 0 , ( x. 2 ) / ( x. 1 , x. 2 ) ) ) '&' ( x. 2 , ( x. 2 ) / ( x. 0 , x. 2 ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( for i being Element of NAT st i in dom p holds ( ( p | ( n + 1 ) ) . i = ( p | n ) . i ) & ( ( p | n ) . i = ( p | n ) . i ) & ( p | ( n + 1 ) ) . i = ( p | n ) . i )
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in of the non empty set & ( the partial F of f ) . ( k + 1 ) is Element of the carrier of a & ( the partial F of f ) . ( k + 1 ) is Element of the carrier of a & ( the Subset of f ) . ( k + 1 ) is Element of the carrier of a ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( a := b ) , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( a := b ) , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) , k ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) * u .= ( ( - f ) *' ) * ( - 1_ F_Complex ) .= ( ( - f ) *' ) * ( - 1_ F_Complex ) ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( 1 - g ) ) .= ( f / g ) /. c * ( ( 1 - g ) * ( 1 - g ) ) .= ( f / g ) /. c ;
len Cf - len ( <* V *> /. 1 ) = len Cf -' len ( <* V *> /. 1 ) .= len ( Cf ) - len ( <* V *> /. 1 ) .= len ( ( f | 1 ) ^ <* V *> ) .= len ( f | 1 ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= dom ( r (#) f ) .= X /\ X .= dom ( r (#) f ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and for n st n < k holds f " { f . n } = { n } and f . n = n + 1 and f . n = n + 1 ;
consider vs be Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and E7 . ( A \/ B ) = Prob ( vs , D ) and E7 . ( A \/ B ) = Prob ( vs , D ) and E7 . ( c \/ d ) = Prob ( vs , D ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] and Q [ y ] ;
assume that A c= Z and f = ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( #Z 2 ) ) ) ) ) `| Z ) = f ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q2 + len Seq q2 & len Seq q2 = len q1 + len Seq q2 & len Seq q1 = len q1 + len q2 & len Seq q1 = len q2 + len q2 & len Seq q2 = len q2 + len q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G2 , G3 and g is Morphism of G1 , G3 and g is Morphism of G2 , G3 and f is Morphism of G2 , G3 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 ;
func - f -> PartFunc of C , V means : Def6 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c * ( - f /. c ) & for c st c in dom it holds it /. c = - f /. c * ( - f /. c ) ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for H holds union ( ( union L ) | [: a , b :] ) |= H iff L . a in ( union L ) | [: a , b :] ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i / n ) * n and for n1 being Nat , n2 being Nat st n1 <> 0 & n2 = 0 & n2 = n & n <= n2 holds ( n / ( n1 + n2 ) ) * ( n / ( n1 + n2 ) ) <= ( n / ( n1 + n2 ) ) * ( n / ( n1 + n2 ) ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( f1 + f2 ) ) `| Z ) and for x st x in Z holds ( ( ( arccot * f1 ) + ( arccot * f2 ) ) `| Z ) . x = - 1 / ( x + x ^2 ) and for x st x in Z holds ( ( ( 1 / 2 ) (#) ( ( #Z * f1 ) + ( arccot * f2 ) ) ) `| Z ) . x = 1 / ( x - x ^2 ) ) ;
cell ( G1 , i1 -' 1 , j2 ) \ ( ( Y -' 1 ) * ( ( A -' 1 ) -' 1 ) + ( A -' 2 ) ) c= BDD L~ f1 \/ L~ ( ( Y -' 1 ) -' 1 ) ) \/ BDD L~ f2 ;
ex Q1 being open Subset of [: X , Y :] st s = Q1 & ex \frac F being Subset-Family of [: Y , X :] , Q1 being Subset-Family of [: Y , X :] st [: F , Q1 :] c= F & ( for x being Point of Y , Q being Subset of [: Y , X :] st x in Q holds x in Q & Q is finite ) & ( ex x being Point of Y st x in Q & x in Q ) & ( ex y being Subset of Y st y in Q & y in Q & y in Q & x in Q & y in Q & y in Q & y in Q & y in Q & y in Q & y in Q & x in Q & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q
gcd ( A9 , ( 1. ( A ) ) , ( 1. ( A ) ) , ( 1. ( A ) ) ) = 1. ( R ) & gcd ( A9 , ( 1. ( A ) ) , ( 1. ( A ) ) ) = 1. ( R ) ;
R8 = ( ( the Subset of ( the carrier of s2 ) ) ) . ( m2 + 1 ) .= ( the InternalRel of ( s2 ) ) . m2 .= ( the InternalRel of ( s2 ) ) . m2 .= ( the InternalRel of ( s2 ) ) . m2 .= ( the InternalRel of ( s2 ) ) . m2 ;
CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= CurInstr ( P3 , s3 ) .= CurInstr ( P3 , s3 ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 } \/ ( LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 } \/ { p2 } ;
func not the still of f -> Subset of the Sorts of Al means : - ex a , b st a in dom f & b in dom f & a = f . a & a in the still of f & b = f . b & a <> b & a in dom f & b in the still of f & a <> b ;
for a , b being Element of F_Complex st |. a .| > |. b .| for f being Polynomial of F_Complex st f is \cup { b } for a being Element of F_Complex st a >= 1 holds f * ( - b ) is \cup { b } holds a * ( - b ) is \cup of { b }
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) holds j < len G & 1 <= j & j + 1 <= width G ;
assume that C1 , C2 are_`2 and for f , g being State of C1 , s1 , s2 being State of C2 , f being State of C1 , s2 being State of C2 st s1 = s2 holds s1 * f is stable & s2 * f is stable & s1 * g is stable & s2 * f is stable & s1 * g is stable & s2 * g is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) .|| .= ||. ( f | X ) .|| .= ||. ( f | X ) .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B in F & A <> B & A misses B holds card F = card G & card G = card F & card G = card G & card G = card G & card G = card G & card G = card G & card G = card G & card G = card G & card G = card G & card G = card G & card G = card ( card G = card ( card G & card G = card G & card F = card G & card G = card G & card G & card G = card G & card F = card G & card G = card G & card G = card G & card G = card G & card
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( \mathop { \rm *> n ) - i |^ s ) = i |^ ( ( s + k ) - i |^ s ) .= i |^ ( s * ( i |^ k ) - i |^ s ) .= i |^ ( s * ( i |^ k ) - i |^ s ) .= i |^ ( ( s * ( i |^ k ) - 1 ) ) - 1 .= i |^ ( ( s * ( i |^ k ) - 1 ) ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( for q st q in rng F holds ( q . ( len q ) = v1 ) & ( for i st i in dom q holds q . ( len q ) = v2 ) & ( for i st i in dom q holds q . i = v2 ) & ( not i in rng q implies q . ( len q ) = v2 ) ;
defpred P [ Element of NAT ] means $1 <= len I implies ( g g = g . ( len I ) ) & ( g = g . ( len I ) implies ( g = g . ( len I ) + $1 ) ) & ( g = g . ( len I + $1 ) ) implies g = ( g . ( len I + $1 ) ) & ( g . ( len I + $1 ) ) = g . ( len I + $1 ) ) ;
for A , B being square Matrix of n , REAL for n , m being Nat holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in I & s . i = b * a & a in I & b in I & b in I & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX means : Def6 : |( Re x , Re y )| = |( Re x , Re y )| - ( Im y ) * |( Im x , Im y )| + ( Im y ) * |( Im y , Im y )| ;
consider g2 be FinSequence of FF such that g2 is continuous and rng g2 c= A and g2 . 1 = x1 and g2 . len g2 = x2 and for k st k in dom g2 & k <> 1 holds g2 . k = F ( k ) and g2 . 1 = F ( k ) and g2 . len g2 = G ( k ) ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n2 , n3 , n2
q `1 * a <= q `1 & - q `1 * a <= q `1 & - q `1 * a <= q `1 * a or q `1 * a >= q `1 & - q `1 * a <= q `1 & - q `1 * a <= q `1 * a or q `1 >= q `1 & q `1 <= - q `1 * a or q `1 >= q `1 & q `1 <= - q `1 * a & q `1 <= - q `1 * a ;
( F . ( p . len p ) ) = ( F . ( p . len p ) ) .= ( F . ( p . len p ) ) .= ( ( F . ( p . len p ) ) ) * ( ( F . ( p . len p ) ) ) .= ( ( F . ( p . len p ) ) * ( ( F . ( p . len p ) ) ) ) * ( ( F . ( p . len p ) ) ) .= ( ( F . ( p . 1 ) ) * ( ( F . 1 ) ) * ( ( F . 1 ) * ( F . 1 ) ) * ( ( F . 1 ) * ( ( F . 1 ) * ( F . 1 ) ) .= ( F . 1 ) * ( F . 1 ) ) * ( ( F . 1 ) .= ( ( F . 1 ) * ( ( F .
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( k1 --> SubFrom ( a , intloc 0 ) ) ) ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a *> ^ <* a
consider B9 being Subset of B1 , y-1 being Function of B1 , A1 such that B9 is finite and D" = the carrier of B1 and D" = the carrier of B2 and D" = the carrier of B1 and D8 = the carrier of B2 and D8 = the carrier of B1 and B8 = the carrier of B2 and B8 = the carrier of B2 ;
v2 . b2 = ( ( curry F2 ) * ( ( curry F2 ) * ( ( id B ) . b2 ) ) ) . b2 .= ( ( curry F2 ) * ( ( ( ( ( id B ) * ( id B ) ) . b2 ) ) ) ) . b2 .= ( ( ( ( ( ( ( ( ( id B ) * ( id B ) ) * ( id B ) ) * ( id B ) ) ) * ( id B ) ) ) ) . b2 .= F2 . b2 ;
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < K holds |. h .| " * ||. ( R + R1 ) /. h .|| < ( |. h .| " * ||. ( R2 + R1 ) /. h .|| ) * ||. ( R2 + R1 ) /. h .|| ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE p2 , q , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p1 , p3 , P , p1 , p2 , p2 & LE p2 , p3 , P , p1 , p2 , p2 , p2 , p2 , P , p1 , p2 , p2 & LE p1 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p3 , p3 , p3 , p3 , p3 , p1 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p3 ,
( ( - x ) .|. y ) = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) *
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 * ( p `1 ) ^2 + ( p `2 / p `1 ) ^2 * ( p `2 ) ^2 ;
( ( ( U - T ) * ( W7 - T ) ) * ( thesis - 1 ) ) = ( ( ( U - T ) * ( W7 - T ) ) * ( p - 1 ) ) * ( ( ( U - T ) * ( p - 1 ) ) * ( p - 1 ) ) .= ( ( U - T ) * ( p - 1 ) ) * ( p - 1 ) ) * ( p - 1 ) .= ( ( U - 1 ) * ( p - 1 ) * ( p - 1 ) ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) * ( p - 1 ) + ( ( ( ( ( p - 1 ) * ( ( p - 1 ) ) * ( ( p - 1 ) * ( p -
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def6 : dom it = ( - h ) * f & for x st x in dom it holds it . x = ( - h ) * f . x + ( - h ) * f . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in variables_in H and not x in Free H and ( not x in Free H \ { x } or x in Free H \ { y } ) and not x in Free H and not x in Free H and not x in Free H and not x in Free H and not x in Free H \ { y } ) and not x in Free H \ { y } ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( P [ $1 ] implies $2 = p |^ $1 & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def6 : for A being Subset of X holds A in it iff for W being Subset of X st W c= X & W c= X \ A holds C . W <= C . ( W \/ Z ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( ( P ) ) .: Q ) ) & lower_bound ( ( dist ( P ) ) .: Q ) = ( ( dist ( P ) ) ) ^2 + ( ( dist ( P ) ) ^2 ) ) ^2 ;
rng ( F | ( [: S , S :] ) ) = {} or rng ( F | ( [: S , S :] ) ) = { 1 } or rng ( F | ( [: S , S :] ) ) = { 2 } or rng ( F | ( [: S , S :] ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " .= ( f . i ) " ( rng f ) .= ( f . i ) " ( rng f ) .= ( f . i ) " ( rng f ) .= ( f . i ) " ( rng f ) .= ( f . i ) " ( rng f ) ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/ P2 \/
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 - 1 / ( ( p2 `1 ) ) ^2 + ( p2 `2 ) ^2 / ( ( p2 `1 ) ) ^2 ) , ( p2 `2 ) ^2 / ( ( p2 `1 ) ) ^2 / ( ( p2 `2 ) ^2 ) ]| ;
( ( proj ( a , X ) ) " ) . x = ( ( proj ( a , X ) qua Function ) qua Function ) . x .= ( ( ( proj ( a , X ) qua Function ) ) . x ) " .= ( ( ( ( proj ( a , X ) ) " ) * ( ( proj ( a , X ) ) " ) ) . x ) .= ( ( - a ) * ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( -
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Point of T st r in B & A misses B for p being Point of T st p in ( Element G ) \ { r } & r in ( ( NAT G ) \ { r } ) holds p in ( ( ( NAT G ) \ { r } ) \/ ( ( NAT G ) \ { r } ) )
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G1 is strict Subgroup of G & G2 is strict Subgroup of G holds G1 is strict Subgroup of G & G2 is strict Subgroup of G
for x st x in Z holds ( ( ( #Z 2 ) * ( arccot - arccot ) ) `| Z ) . x = ( ( #Z 2 ) * ( arccot - arccot ) ) `| Z ) . x / ( ( ( #Z 2 ) * ( arccot - arccot ) ) . x ) ^2
synonym f is_right x0 means : for x st x in dom ( f /* a ) & x in dom f & for a st a in ]. x0 , x0 + r .[ holds f . x - f . x0 < ( f /* a ) . x - f . x0 ;
then X1 , X2 are_separated & Y1 misses Y2 or ex Y1 , Y2 being non empty SubSpace of X st Y1 , Y2 are_separated & Y1 is SubSpace of Y2 & Y2 is SubSpace of Y2 & Y2 is SubSpace of Y1 & Y2 is SubSpace of Y2 & Y1 is SubSpace of Y2 & Y2 is SubSpace of Y2 & Y2 is SubSpace of Y2 & Y2 is SubSpace of Y2 & Y2 is SubSpace of Y2 & Y2 is SubSpace of X1 & Y2 is SubSpace of X2 ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) >= ( ( p2 `1 ) * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ;
( ( 1 / t1 ) (#) ||. f1 .|| ) to_power n = ( ( 1 / t1 ) (#) ||. g1 .|| ) to_power m & ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n = ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n & ( ( 1 / t2 ) (#) ||. g1 .|| ) to_power n = ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n ) to_power n ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin / cos ) ) . x and x in dom ( ( - 1 / 2 ) (#) ( cos / cos ) ) and for x st x in dom ( ( - 1 / 2 ) (#) ( sin / cos ) ) holds ( ( - 1 / 2 ) (#) ( sin / cos ) ) . x = ( - 1 / 2 ) * ( sin / cos ) . x ;
consider X-23 being Subset of [: Y , Y :] , X-22 being Subset of [: X , Y :] such that Xf1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y2 is open and Y1 is open and Y1 is open and Y2 is open and Y1 is open and Y1 is open and Y1 is open and Y2 is open ;
card S . n = card { |[ d , Y |^ 3 + ( a * d ) / b , b * d ]| where d is Element of GF ( p ) : [ d , Y ] in R & [ d , Y ] in R } .= { 1 } \/ { 1 } .= { 1 } \/ { 1 } .= { 1 } \/ { 1 } .= { 1 } \/ { 1 } .= { 1 } \/ { 1 } ;
( ( W-bound D - W-bound D ) / 2 ) * ( i - 2 ) * ( i - 2 ) = ( E-bound D - ( W-bound D - ( W-bound D - ( W-bound D - ( W-bound D - ( W-bound D - ( W-bound D - ( E-bound D - 2 ) / 2 ) ) / 2 ) ) ) * ( i - 2 ) .= ( E-bound D - ( E-bound D - ( E-bound D - ( E-bound D - / 2 ) ) / 2 ) ) * ( i - 2 ) ) ;